Calculus Examples

$f (x) = \frac{e^{x}}{3 + e^{x}}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x}$ and $g (x) = 3 + e^{x}$ .

$\frac{(3 + e^{x}) \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [3 + e^{x}]}{{(3 + e^{x})}^{2}}$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{(3 + e^{x}) e^{x} - e^{x} \frac{d}{d x} [3 + e^{x}]}{{(3 + e^{x})}^{2}}$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

By the Sum Rule, the derivative of $3 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [e^{x}]$ .

$\frac{(3 + e^{x}) e^{x} - e^{x} (\frac{d}{d x} [3] + \frac{d}{d x} [e^{x}])}{{(3 + e^{x})}^{2}}$

Step 1.3.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{(3 + e^{x}) e^{x} - e^{x} (0 + \frac{d}{d x} [e^{x}])}{{(3 + e^{x})}^{2}}$

Step 1.3.3

Add $0$ and $\frac{d}{d x} [e^{x}]$ .

$\frac{(3 + e^{x}) e^{x} - e^{x} \frac{d}{d x} [e^{x}]}{{(3 + e^{x})}^{2}}$

Step 1.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{(3 + e^{x}) e^{x} - e^{x} e^{x}}{{(3 + e^{x})}^{2}}$

Step 1.5

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

Tap for more steps...

Step 1.5.1

Move $e^{x}$ .

$\frac{(3 + e^{x}) e^{x} - (e^{x} e^{x})}{{(3 + e^{x})}^{2}}$

Step 1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(3 + e^{x}) e^{x} - e^{x + x}}{{(3 + e^{x})}^{2}}$

Step 1.5.3

Add $x$ and $x$ .

$\frac{(3 + e^{x}) e^{x} - e^{2 x}}{{(3 + e^{x})}^{2}}$

Step 1.6

Simplify.

Tap for more steps...

Step 1.6.1

Apply the distributive property.

$\frac{3 e^{x} + e^{x} e^{x} - e^{2 x}}{{(3 + e^{x})}^{2}}$

Step 1.6.2

Simplify the numerator.

Tap for more steps...

Step 1.6.2.1

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

Tap for more steps...

Step 1.6.2.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 e^{x} + e^{x + x} - e^{2 x}}{{(3 + e^{x})}^{2}}$

Step 1.6.2.1.2

Add $x$ and $x$ .

$\frac{3 e^{x} + e^{2 x} - e^{2 x}}{{(3 + e^{x})}^{2}}$

Step 1.6.2.2

Combine the opposite terms in $3 e^{x} + e^{2 x} - e^{2 x}$ .

Tap for more steps...

Step 1.6.2.2.1

Subtract $e^{2 x}$ from $e^{2 x}$ .

$\frac{3 e^{x} + 0}{{(3 + e^{x})}^{2}}$

Step 1.6.2.2.2

Add $3 e^{x}$ and $0$ .

$\frac{3 e^{x}}{{(3 + e^{x})}^{2}}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 e^{x}}{{(3 + e^{x})}^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{e^{x}}{{(3 + e^{x})}^{2}}]$ .

$f'' (x) = 3 \frac{d}{d x} (\frac{e^{x}}{{(3 + e^{x})}^{2}})$

Step 2.2

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} \frac{d}{d x} (e^{x}) - e^{x} \frac{d}{d x} {(3 + e^{x})}^{2}}{{({(3 + e^{x})}^{2})}^{2}})$

Step 2.3

Multiply the exponents in ${({(3 + e^{x})}^{2})}^{2}$ .

Tap for more steps...

Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} \frac{d}{d x} (e^{x}) - e^{x} \frac{d}{d x} {(3 + e^{x})}^{2}}{{(3 + e^{x})}^{2 \cdot 2}})$

Step 2.3.2

Multiply $2$ by $2$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} \frac{d}{d x} (e^{x}) - e^{x} \frac{d}{d x} {(3 + e^{x})}^{2}}{{(3 + e^{x})}^{4}})$

Step 2.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - e^{x} \frac{d}{d x} {(3 + e^{x})}^{2}}{{(3 + e^{x})}^{4}})$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 + e^{x}$ .

Tap for more steps...

Step 2.5.1

To apply the Chain Rule, set $u$ as $3 + e^{x}$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - e^{x} (\frac{d}{d u} (u^{2}) \frac{d}{d x} (3 + e^{x}))}{{(3 + e^{x})}^{4}})$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - e^{x} (2 u \frac{d}{d x} (3 + e^{x}))}{{(3 + e^{x})}^{4}})$

Step 2.5.3

Replace all occurrences of $u$ with $3 + e^{x}$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - e^{x} (2 (3 + e^{x}) \frac{d}{d x} (3 + e^{x}))}{{(3 + e^{x})}^{4}})$

Step 2.6

Differentiate.

Tap for more steps...

Step 2.6.1

Multiply $2$ by $- 1$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{x} ((3 + e^{x}) \frac{d}{d x} (3 + e^{x}))}{{(3 + e^{x})}^{4}})$

Step 2.6.2

By the Sum Rule, the derivative of $3 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [e^{x}]$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{x} ((3 + e^{x}) (\frac{d}{d x} (3) + \frac{d}{d x} (e^{x})))}{{(3 + e^{x})}^{4}})$

Step 2.6.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{x} ((3 + e^{x}) (0 + \frac{d}{d x} (e^{x})))}{{(3 + e^{x})}^{4}})$

Step 2.6.4

Add $0$ and $\frac{d}{d x} [e^{x}]$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{x} ((3 + e^{x}) \frac{d}{d x} (e^{x}))}{{(3 + e^{x})}^{4}})$

Step 2.7

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{x} ((3 + e^{x}) e^{x})}{{(3 + e^{x})}^{4}})$

Step 2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{x + x} (3 + e^{x})}{{(3 + e^{x})}^{4}})$

Step 2.9

Add $x$ and $x$ .

$f'' (x) = 3 (\frac{{(3 + e^{x})}^{2} e^{x} - 2 e^{2 x} (3 + e^{x})}{{(3 + e^{x})}^{4}})$

Step 2.10

Factor $3 + e^{x}$ out of ${(3 + e^{x})}^{2} e^{x} - 2 e^{2 x} (3 + e^{x})$ .

Tap for more steps...

Step 2.10.1

Factor $3 + e^{x}$ out of ${(3 + e^{x})}^{2} e^{x}$ .

$f'' (x) = 3 (\frac{(3 + e^{x}) ((3 + e^{x}) e^{x}) - 2 e^{2 x} (3 + e^{x})}{{(3 + e^{x})}^{4}})$

Step 2.10.2

Factor $3 + e^{x}$ out of $- 2 e^{2 x} (3 + e^{x})$ .

$f'' (x) = 3 (\frac{(3 + e^{x}) ((3 + e^{x}) e^{x}) + (3 + e^{x}) (- 2 e^{2 x})}{{(3 + e^{x})}^{4}})$

Step 2.10.3

Factor $3 + e^{x}$ out of $(3 + e^{x}) ((3 + e^{x}) e^{x}) + (3 + e^{x}) (- 2 e^{2 x})$ .

$f'' (x) = 3 (\frac{(3 + e^{x}) ((3 + e^{x}) e^{x} - 2 e^{2 x})}{{(3 + e^{x})}^{4}})$

Step 2.11

Cancel the common factors.

Tap for more steps...

Step 2.11.1

Factor $3 + e^{x}$ out of ${(3 + e^{x})}^{4}$ .

$f'' (x) = 3 (\frac{(3 + e^{x}) ((3 + e^{x}) e^{x} - 2 e^{2 x})}{(3 + e^{x}) {(3 + e^{x})}^{3}})$

Step 2.11.2

Cancel the common factor.

$f'' (x) = 3 (\frac{(3 + e^{x}) ((3 + e^{x}) e^{x} - 2 e^{2 x})}{(3 + e^{x}) {(3 + e^{x})}^{3}})$

Step 2.11.3

Rewrite the expression.

$f'' (x) = 3 (\frac{(3 + e^{x}) e^{x} - 2 e^{2 x}}{{(3 + e^{x})}^{3}})$

Step 2.12

Combine $3$ and $\frac{(3 + e^{x}) e^{x} - 2 e^{2 x}}{{(3 + e^{x})}^{3}}$ .

$f'' (x) = \frac{3 ((3 + e^{x}) e^{x} - 2 e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13

Simplify.

Tap for more steps...

Step 2.13.1

Apply the distributive property.

$f'' (x) = \frac{3 (3 e^{x} + e^{x} e^{x} - 2 e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.2

Apply the distributive property.

$f'' (x) = \frac{3 (3 e^{x}) + 3 (e^{x} e^{x}) + 3 (- 2 e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.3

Simplify the numerator.

Tap for more steps...

Step 2.13.3.1

Simplify each term.

Tap for more steps...

Step 2.13.3.1.1

Multiply $3$ by $3$ .

$f'' (x) = \frac{9 e^{x} + 3 (e^{x} e^{x}) + 3 (- 2 e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.3.1.2

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

Tap for more steps...

Step 2.13.3.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{9 e^{x} + 3 e^{x + x} + 3 (- 2 e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.3.1.2.2

Add $x$ and $x$ .

$f'' (x) = \frac{9 e^{x} + 3 e^{2 x} + 3 (- 2 e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.3.1.3

Multiply $- 2$ by $3$ .

$f'' (x) = \frac{9 e^{x} + 3 e^{2 x} - 6 e^{2 x}}{{(3 + e^{x})}^{3}}$

Step 2.13.3.2

Subtract $6 e^{2 x}$ from $3 e^{2 x}$ .

$f'' (x) = \frac{9 e^{x} - 3 e^{2 x}}{{(3 + e^{x})}^{3}}$

Step 2.13.4

Simplify the numerator.

Tap for more steps...

Step 2.13.4.1

Factor $3$ out of $9 e^{x} - 3 e^{2 x}$ .

Tap for more steps...

Step 2.13.4.1.1

Factor $3$ out of $9 e^{x}$ .

$f'' (x) = \frac{3 (3 e^{x}) - 3 e^{2 x}}{{(3 + e^{x})}^{3}}$

Step 2.13.4.1.2

Factor $3$ out of $- 3 e^{2 x}$ .

$f'' (x) = \frac{3 (3 e^{x}) + 3 (- e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.4.1.3

Factor $3$ out of $3 (3 e^{x}) + 3 (- e^{2 x})$ .

$f'' (x) = \frac{3 (3 e^{x} - e^{2 x})}{{(3 + e^{x})}^{3}}$

Step 2.13.4.2

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

$f'' (x) = \frac{3 (3 e^{x} - {(e^{x})}^{2})}{{(3 + e^{x})}^{3}}$

Step 2.13.4.3

Let $u = e^{x}$ . Substitute $u$ for all occurrences of $e^{x}$ .

$f'' (x) = \frac{3 (3 u - u^{2})}{{(3 + e^{x})}^{3}}$

Step 2.13.4.4

Factor $u$ out of $3 u - u^{2}$ .

Tap for more steps...

Step 2.13.4.4.1

Factor $u$ out of $3 u$ .

$f'' (x) = \frac{3 (u \cdot 3 - u^{2})}{{(3 + e^{x})}^{3}}$

Step 2.13.4.4.2

Factor $u$ out of $- u^{2}$ .

$f'' (x) = \frac{3 (u \cdot 3 + u (- u))}{{(3 + e^{x})}^{3}}$

Step 2.13.4.4.3

Factor $u$ out of $u \cdot 3 + u (- u)$ .

$f'' (x) = \frac{3 (u (3 - u))}{{(3 + e^{x})}^{3}}$

Step 2.13.4.5

Replace all occurrences of $u$ with $e^{x}$ .

$f'' (x) = \frac{3 e^{x} (3 - e^{x})}{{(3 + e^{x})}^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{3 e^{x}}{{(3 + e^{x})}^{2}} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6